generalized Farkas lemma
The more abstract version of Farkas’ Lemma is useful for understanding the essence of the usual version of the lemma proven for matrices, and of course, for solving optimization problems in infinite-dimensional spaces.
0.1 Formal statements
Given , and a weakly-closed convex cone , the following are equivalent:
The equivalence of conditions (a) and (c) is a fundamental property of the anti-cone, while condition (b) is merely a rephrasal of condition (c). ∎
Let be a real vector space, and be a subspace of linear functionals on that separate points. Impose on the weak-* topology generated by .
Given a functional , and a weak-* closed convex cone , the following are equivalent:
Make the substitutions , , and in Theorem 1. ∎
Let and be real vector spaces, with corresponding spaces of linear functionals and that separate points. Have and generate the weak topology for and respectively.
Given , a linear mapping , and a subset such that is a weakly-closed convex cone, the following are equivalent:
The linear equation has a solution .
If satisfies for all , then .
If satisfies (anti-cone of with respect to ), then .
Here denotes the pullback, restricted to and , defined by .
Make the substitutions , , and in Theorem 1. Condition (c) is a rephrasal of condition (b). ∎
|Title||generalized Farkas lemma|
|Date of creation||2013-03-22 17:20:53|
|Last modified on||2013-03-22 17:20:53|
|Last modified by||stevecheng (10074)|
|Synonym||Farkas lemma for topological vector spaces|
|Synonym||generalized Farkas theorem|